Universal microscopic correlation functions for products of independent Ginibre matrices

Akemann G, Burda Z (2012)
J. Phys. A: Math. Theor. 45(46): 465201.

Zeitschriftenaufsatz | Veröffentlicht | Englisch
 
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Abstract / Bemerkung
We consider the product of n complex non-Hermitian, independent randommatrices, each of size NxN with independent identically distributed Gaussianentries (Ginibre matrices). The joint probability distribution of the complexeigenvalues of the product matrix is found to be given by a determinantal pointprocess as in the case of a single Ginibre matrix, but with a more complicatedweight given by a Meijer G-function depending on n. Using the method oforthogonal polynomials we compute all eigenvalue density correlation functionsexactly for finite N and fixed n. They are given by the determinant of thecorresponding kernel which we construct explicitly. In the large-N limit atfixed n we first determine the microscopic correlation functions in the bulkand at the edge of the spectrum. After unfolding they are identical to that ofthe Ginibre ensemble with n=1 and thus universal. In contrast the microscopiccorrelations we find at the origin differ for each n and generalise the knownBessel-law in the complex plane for n=2 to a new hypergeometric kernel 0_F_n-1.
Erscheinungsjahr
2012
Zeitschriftentitel
J. Phys. A: Math. Theor.
Band
45
Ausgabe
46
Art.-Nr.
465201
ISSN
1751-8113
eISSN
1751-8121
Page URI
https://pub.uni-bielefeld.de/record/2518752

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Akemann G, Burda Z. Universal microscopic correlation functions for products of independent Ginibre matrices. J. Phys. A: Math. Theor. 2012;45(46): 465201.
Akemann, G., & Burda, Z. (2012). Universal microscopic correlation functions for products of independent Ginibre matrices. J. Phys. A: Math. Theor., 45(46), 465201. doi:10.1088/1751-8113/45/46/465201
Akemann, Gernot, and Burda, Zdzislaw. 2012. “Universal microscopic correlation functions for products of independent Ginibre matrices”. J. Phys. A: Math. Theor. 45 (46): 465201.
Akemann, G., and Burda, Z. (2012). Universal microscopic correlation functions for products of independent Ginibre matrices. J. Phys. A: Math. Theor. 45:465201.
Akemann, G., & Burda, Z., 2012. Universal microscopic correlation functions for products of independent Ginibre matrices. J. Phys. A: Math. Theor., 45(46): 465201.
G. Akemann and Z. Burda, “Universal microscopic correlation functions for products of independent Ginibre matrices”, J. Phys. A: Math. Theor., vol. 45, 2012, : 465201.
Akemann, G., Burda, Z.: Universal microscopic correlation functions for products of independent Ginibre matrices. J. Phys. A: Math. Theor. 45, : 465201 (2012).
Akemann, Gernot, and Burda, Zdzislaw. “Universal microscopic correlation functions for products of independent Ginibre matrices”. J. Phys. A: Math. Theor. 45.46 (2012): 465201.
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